Nonlinear Combustion Instability in Liquid-Propellant .Nonlinear Combustion Instability in...
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Nonlinear Combustion Instability in
Liquid-Propellant Rocket Motors
Samuel Z. Burstein and Wallace Chinitz
Second Quarterly Report
to . -
Jet Propulsion Laboratory
January 30, 1968
JPL COMTBACT 951946
"This work was performed for the Jet Propulsion Laboratory, California Institute of Technology, as sponsored by the National Aeronautics and Space Administration under Contract NAS7-100."
"This report contains information prepared
by Mathematical Applications Group, Inc.
(MAGI) under JPL subcontract. Its content
is not necessarily endorsed by the Jet
Propulsion Laboratory, California Institute
of Technology, or the National Aeronautics
and Space Administration."
TABLE OF CONTENTS
INTRODUCTION ...................................... 1
DROPLET EVAPORATION AND COMBUSTION ANALYSES ....... 2 A . The Rate of the N2H4/N204 Reaction ........... 2 B . The Diffusion-Controlled Flame Analysis ..... 11
. FLUID DYNAMIC MODEL .............................. 17 A . First-Row Mesh Point Calculation ............ 17 B . An Exact Solution-Comparison with
Machine Experiment ......................... 19 C . A Finite Amplitude Calculation of a
Rotating Wave .............................. 21 D . Convergence of the Difference Equations ..... 2 3 E . Simple Forcing Function ..................... 2 6 F . Program COMB Status ......................... 28
During the second quarter of the contract there has been
significant progress in the construction of a mathematical model
describing the dynamics of a rotating finite-amplitude wave in a
cylindrical combustion chamber. This quarterly report describes
some of the details of this activity.
The first section describes the theory of finite rate proc-
esses in the combustion of hydrazine (NzHq)/nitrogen tetroxide
for the modified surface flame model of combustion. Also discussed
the results will be used to predict kinetic parameters
in this section are some results obtained from the diffusion flame
The second section is involved with the fluid dynamic model.
A new treatment for the calculation of special mesh points in
COMB is presented. Most important, however, the results of some
time dependent computations are described in which a finite am-
plitude pressure wave is allowed to rotate and steepen in a
cylindrical chamber. The reader should refer to reference 1 for
background material and additional references (especially reference
2 0 ) .
The authors would like to acknowledge the significant contri-
bution of Harold Schechter in the organization, analysis and
programming of the digital programs being developed in this
I. THE DROPLET EVAPORATION AND COMBUSTION ANALYSES
A . The Rate of the N2H4/N204 Reaction
It will be recalled (Ref. 1) that the species conservation
equation including finite rate chemical reaction can be written:
dYi - d 2 dYi 2 N F yNo (1) a- ( n -) = - iin bYF dn 0
where a = the dimensionless mass burning rate
Yi = mass fraction of species i
n = radius/droplet radius
Mi = molecular weight of species i
p = mass density
NF,N, = reaction orders 2
b = rDk/D
k = Arrhenius reaction rate constant = A(T)exp(-E/RT)
D = diffusion coefficient
It was also shown (Ref. 1) that by neglecting the energy
transport due to concentration gradients (the Dufour effect) and
assuming the Lewis number to be unity, the energy equation takes
on the same form as equation (1) above:
2 NF NO a- - - - i q b Y F (,,* dt -
dt dn dn -1 -
In order to apply Peskin's (Ref. 2) modified flame surface
analysis, realistic estimates must be obtained for the reaction
rate constant, k = A (T) exp (-E/RT) , and the reaction orders, NF and No, corresponding to the N2H*/N204 reaction. These can
be obtained from a one-dimensional, finite-rate analysis of the
oxidation process, using the simplified chemical kinetic mechanism
deduced from Sawyer, and discussed in Ref. 1. This method is
formulated in the following manner.
As mentioned above, the flow equations are written for one-
dimensional, constant pressure, inviscid flow along streamtubes,
following additional assumptions:
Negligible diffusion normal to the flow direction,
leading to a uniform mixture of (gaseous) species
at any streamwise point. Therefore, the oxidation
is taken to be 'reaction-controlled'.
The flow is steady and adiabatic.
The initial concentrations of the oxidizing species
are obtained in accordance with the following
reaction equation and with the assumption of
N2O4 + a N204+bN02 + cNO + d02 ( 3 ) The method for obtaining the equilibrium species concentrations
is detailed in Appendix A.
With these assumptions, the relevant equations are:
dh = ; h = constant dt
where h includes sensible - and chemical enthalpy.
dYi = ri Pdt (5)
Equation of State
P = p M RT
h = C Yihi i
M = C Yi (i F)
The enthalpy terms are taken to be linear in form: -
hi = Ai + c Pi
where the reference temperature was chosen to be 1000'K. Values
for Ai and E
chemical kinetic mechanism and t h e species generation terms, ri,
are in Tables I and 11, respectively. The reaction rate constants
are in Table I.
were obtained from the data of Reference 5. The Pi
It should be emphasized that this premixed, homogeneous gas-
phase analysis is designed to supply an appropriate representation
of the overall (global) oxidation reaction rate, for use in equa-
tions (1) and ( 2 ) . It is not, of course, a representation of the
processes occurring around the fuel drop, where a counter-flow
diffusion flame exists, as described by equations (1) and ( 2 ) .
Results for two values of the ambient pressure are shown in
Figures 1 and 2 . At one atmosphere (Figure 11, and an initial
temperature of 1500'K, the reaction is seen to commence at about
seconds, continue smoothly until about seconds after
. which a rapid rise in temperature is noted. If the calculations
were continued, a rapid leveling off to the adiabatic flame tem-
perature would be noted. In this case, the ignition delay time
can be estimated to be seconds.
At the lower initial temperature (Ti = 1000K), the ignition
delay time is seen to increase in Figure 1 to about S X ~ O - ~ seconds.
When the pressure is increased to 20 atmospheres (Figure 2 ) ,
a somewhat different behavior is noted. At both initial temper-
atures, the reaction is seen to commence quite early (at about
10 seconds for Ti = 1500K, and before lo seconds for Ti =
1000K). However, a pause in the reaction then occurs and, for
the case of Ti = 1500K, between 5x10- and 5x10-* seconds little
temperature change is noted. After 5x10-* seconds, a smooth and,
ultimately, rapid rise in temperature is noted.
The reason for this behavior can be explained by reference
to Figure 3 and Table 11. In Figure 3, it is seen that NO2 dis-
appears rapidly during this initial temperature rise, with a
corresponding decrease in the concentration of hydrazine. This
same behavior was not noted f o r the case of p = 1 atm because
reaction number 2 in Table I1 is second-order and, hence, pre-
dominates at higher pressures. An accompanying increase of NO,
as a result of reaction 2, can be seen in Figure 3 as well.
After the depletion of the N02, the N2H4/NO and N2H4/O2
oxidation reactions take over and proceed as first-order reactions
(Table 11), leading to a second delay period, and a subsequent
rapid reaction at about seconds. The behavior of NH3 and
H 2 is shown in Figure 4 . It can be seen that they rise in concen-
tration initially, as a consequence of the hydrazine decomposition
reaction (reaction 1, Table 11), then rapidly disappear when the
main reaction occurs.
It is clear then, that the 'ignition delay time' is difficult
to define in the pressure regime where this two-step reaction
occurs. It can be anticipated that the traditional plot of igni-
tion delay as a function of reciprocal temperature will not produce
a straight line since the rate-controlling reactions are different
at different pressures. This will lead to the requirement for a
different 'global' reaction rate constant for the high and low
pressure regimes. Current efforts are directed toward obtaining
sufficient computer runs so that the chemical kinetic parameters
can be determined in both pressure regimes.
B. The Diffusion-Controlled Flame Analysis
The diffusion-controlled flame analysis which was described
in Reference 1 has been coupled to the basic computer program
COMB. In this section, we compare the results obtained from t